scholarly journals On the von Neumann entropy of certain quantum walks subject to decoherence

2010 ◽  
Vol 20 (6) ◽  
pp. 1099-1115 ◽  
Author(s):  
CHAOBIN LIU ◽  
NELSON PETULANTE
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Total Density ◽  
Projective Measurement ◽  
Time Step ◽  
Von Neumann ◽  
Bipartite Quantum System ◽  
Maximum Value ◽  
Time Quantum

In this paper, we consider a discrete-time quantum walk on the N-cycle governed by the condition that at every time step of the walk, the option persists, with probability p, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems corresponding to the space of the coin and the space of the walker must eventually diminish to zero. Put plainly, any level of decoherence greater than zero forces the system to become completely ‘disentangled’ eventually.

Entanglement for quantum walks on the line

2011 ◽  
Vol 11 (9&10) ◽  
pp. 855-866
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Initial State ◽  
Von Neumann ◽  
Time Quantum ◽  
The One ◽  
Path Counting

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

scholarly journals ON THE VON NEUMANN AND SHANNON ENTROPIES FOR QUANTUM WALKS ON Z2

2012 ◽  
Vol 10 (02) ◽  
pp. 1250020 ◽  
Author(s):  
CLEMENT AMPADU
Keyword(s):  
Discrete Time ◽  
Shannon Entropy ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Asymptotic Behaviors ◽  
Von Neumann ◽  
Time Quantum

We give asymptotic behaviors of the von Neumann entropy and the Shannon entropy of discrete-time time quantum walks on Z2.

scholarly journals Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 586 ◽  
Author(s):  
Xin Wang ◽  
Yi Zhang ◽  
Kai Lu ◽  
Xiaoping Wang ◽  
Kai Liu
Keyword(s):  
Polynomial Time ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Walk ◽  
Isomorphism Problem ◽  
Quantum Walks ◽  
Regular Graphs ◽  
Structure Preserving ◽  
Topological Structures ◽  
Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Discrete Time ◽  
Quantum Teleportation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
One Dimensional ◽  
Time Quantum ◽  
Infinite Line ◽  
The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

2006 ◽  
Vol 09 (02) ◽  
pp. 287-297 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Probability Theory ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Homogeneous Tree ◽  
One Dimensional ◽  
Time Quantum ◽  
New Type ◽  
The One

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

scholarly journals MIXING OF QUANTUM WALKS ON GENERALIZED HYPERCUBES

2008 ◽  
Vol 06 (06) ◽  
pp. 1135-1148 ◽  
Author(s):  
ANA BEST ◽  
MARKUS KLIEGL ◽  
SHAWN MEAD-GLUCHACKI ◽  
CHRISTINO TAMON
Keyword(s):  
Mixing Time ◽  
Quantum Walk ◽  
Circulant Matrix ◽  
Quantum Walks ◽  
Spectral Structure ◽  
The Status ◽  
Time Quantum ◽  
Hamming Graphs ◽  
The Rich ◽  
The Fourier Transform

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1 on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph [Formula: see text] whose adjacency matrix is I ⊗ Qn + X ⊗ Af, where Af is a [Formula: see text]-circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.

scholarly journals Probing the Eigenstates Thermalization Hypothesis with Many-Particle Quantum Walks on Lattices

2017 ◽  
Vol 24 (02) ◽  
pp. 1750007
Author(s):  
Dibwe Pierrot Musumbu ◽  
Maria Przybylska ◽  
Andrzej J. Maciejewski
Keyword(s):  
Thermal State ◽  
Particle System ◽  
Hamiltonian Dynamics ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Grid Graphs ◽  
Time Quantum ◽  
Counting Statistics ◽  
The Many ◽  
The One

We simulate the dynamics of many-particle system of bosons and fermions using discrete time quantum walks on lattices. We present a computational proof of the behaviour of the simulated systems similar to the one observed in Hamiltonian dynamics during quantum thermalization. We record the time evolution of the entropy and the temperature of a specific particle configuration during the entire dynamics and observe how they relax to a state which we call the quantum walk thermal state. This observation is made on two types of lattices while simulating different numbers of particles walking on two grid graphs with 25 vertices. In each case, we observe that the vertices counting statistics, the temperature of the indexed configuration and the dimension of the effective configuration Hilbert space relax simultaneously and remain relaxed for the rest of the many-particle quantum walk.

scholarly journals Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 504
Author(s):  
Ce Wang ◽  
Caishi Wang
Keyword(s):  
Discrete Time ◽  
Quantum Information Processing ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Integer Lattice ◽  
Initial State ◽  
Higher Dimensional ◽  
Time Quantum ◽  
Gauss Type ◽  
Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

Accelerating Stochastic Dynamics Simulation With Continuous-Time Quantum Walks

2016 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Continuous Time ◽  
Stochastic Dynamics ◽  
Diffusion Processes ◽  
Dimensional Space ◽  
Planck Equation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Dynamics Simulation ◽  
Time Quantum

Stochastic diffusion is a general phenomenon observed in various national and engineering systems. It is typically modeled by either stochastic differential equation (SDE) or Fokker-Planck equation (FPE), which are equivalent approaches. Path integral is an accurate and effective method to solve FPEs. Yet, computational efficiency is the common challenge for path integral and other numerical methods, include time and space complexities. Previously, one-dimensional continuous-time quantum walk was used to simulate diffusion. By combining quantum diffusion and random diffusion, the new approach can accelerate the simulation with longer time steps than those in path integral. It was demonstrated that simulation can be dozens or even hundreds of times faster. In this paper, a new generic quantum operator is proposed to simulate drift-diffusion processes in high-dimensional space, which combines quantum walks on graphs with traditional path integral approaches. Probability amplitudes are computed efficiently by spectral analysis. The efficiency of the new method is demonstrated with stochastic resonance problems.

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